A person's height membership function graph is shown next with linguistic values of the degree of membership as very tall, tall, average, short and very short being replaced by 0.85, 0.65, 0.50, 0.45 and 0.15.

4

In traditional logic, statements can be either true or false, and sets can either contain an element or not.

These logic values and set memberships are typically represented with number 1 and 0.

Two fuzzy functions are valid iff the function outputs are ? 0.5 under all possible assignments.

This is like doing EXOR of two binary functions shown in (b) which is the same as union.

Two fuzzy functions are inconsistent iff the function output is ? 0.5 under all possible assignments. Thus, if the output of the two fuzzy functions is lt 0.5 then the two fuzzy functions are inconsistent.

This is like exnor of two binary functions of shown in (a) which is the same as intersection.

22Fuzzy Logic

The concept of fuzzy logic was introduced by L.A Zadeh in a 1965 paper.

Aristotelian concepts have been useful and applicable for many years.

But they present us with certain problems

Cannot express ambiguity

Lack of quantifiers

Cannot handle exceptions

23Traditional Logic Problems

Cannot express ambiguity

Consider the predicate X is tall'.

Providing a specific person we can turn the predicate into a statement.

But what is the exact meaning of the word tall'?

What is tall' to some people is not tall to others.

Lack of quantifiers

Another problem is the lack of being able to express statements such as Most of the goals came in the first half '.

The most' quantifier cannot be expressed in terms of the universal and/or existential quantifiers.

24Traditional Logic Problems

Cannot handle exceptions

Another limitation of traditional predicate logic is expressing things that are sometimes, but not always true.

25Traditional sets

In order to represent a set we use curly brackets .

Within the curly brackets we enclose the names of the items, separating them from each other by commas.

The items within the curly brackets are referred to as the elements of the set.

Example Set of vowels in the English alphabet a,e,i,o,u

When dealing with numerical elements we may replace any number of elements using 3 dots.

Example Set of numbers from 1 to 100 1,2,3,...,100

Set of numbers from 23 to infinity 23,24,25,...

26Traditional sets

Rather than writing the description of a set all the time we can give names to the set.

The general convention is to give sets names in capital letters.

Example

V set of vowels in the English alphabet.

Hence any time we encounter V implies the set a, e, i, o, u.

For finite size sets a diagrammatic representation can be employed which can be used to assist in their understanding.

These are called the cloud diagrams

27Cloud Diagrams 28Set order

The order in which the elements are written down is not important.

Example V a,e,i,o,u u,o,i,e,a a,o,e,u,i

The names of the elements in a set must be unique.

Example

V a,a,e,i,o,u

If two elements are the same then there is no point writing them down twice (waste of effort)

but if different then we must introduce a way to tell them apart.

29Set membership

Given any set, we can test if a certain thing is an element of the set or not.

The Greek symbol, ?, indicates an element is a member of a set.

For example, x?A means that x is an element of the set A.

If an element is not a member of a set, the symbol ? is used, as in ?A.

30Set equality subsets

Two sets A and B are equal, (A B) if every element of A is an element of B and every element of B is an element of A.

A set A is a subset of set B, (A ? B) if every element of A is an element of B.

A set A is a proper subset of set B, (A ? B) if A is a subset of B and the two sets are not equal.

31Set equality subsets

Two sets A and B are disjoint, (A ? b) if and only if their intersection is the empty set.

There are a number of special sets. For instance

Boolean BTrue, False

Natural numbers N0,1,2,3,...

Integer numbers Z...,-3,-2,-1,0,1,2,3,...

Real numbers R

Characters Char

Empty set ? or

The empty set is not to be confused with 0 which is a set which contains the number zero as its only element.

The certainty factor indicates the net belief in the conclusion and premises of a rule based on some evidence.

Certainty factors are hand-crafted by asking potential users questions such as How much do you believe that opening valve x will start a flooding' and How much do you disbelieve that opening valve x will start a flooding'.

The degree of certainty is the difference between the two responses.

129Production Rules

Assuming that the knowledge-base module contains knowledge represented in the the format of production rules the following sections introduce the following

the concept of a production rule

the concept of linguistic variables

the fuzzy inference concept

the concept of fuzzification and how to accomplish the crisp to fuzzy transformation

the concept of defuzzification and how to accomplish the fuzzy to crisp transformation

130Knowledge presentation using production rules

From a philosophical point the concept of knowledge is highly ambiguous and debatable

knowledge-base builders treat knowledge from a narrower point of view.

This way the knowledge is easier to model and understand.

It remains diverse including

rules,

facts,

truths,

reasons,

defaults and

heuristics.

The knowledge engineer needs some technique for capturing what is known about the application.

131Knowledge presentation using production rules

The technique should provide expressive adequacy and notational efficacy.

Knowledge representation is very much under constant research.

Several schemes have been suggested in the literature, namely

semantic nets,

frames and

logic.

Production rules have also been suggested and are the most popular way of representing knowledge.

132Knowledge presentation using production rules

Production rules are small chunks of knowledge expressed in the form of if..then statements.

The left hand side (IF) represents the antecedent or conditional part.

The right hand side (THEN) represents the conclusion or action part.

A number of rules collectively define a modularized know-how system.

The principal use of production rules is in the encoding of empirical associations between incoming patterns of data and actions that the system should perform as a consequence.

The production rules are either expressed by an expert of the field, or derived using induction.

133Fuzzy Logic Control

Fuzzy controller design consist of turning intuitions, and any other information about how to control a system, into set of rules.

These rules can then be applied to the system.

If the rules adequately control the system, the design work is done.

If the rules are inadequate, the way they fail provides information to change the rules.

134Control a Plant

A valve in an internal combustion engine that regulates the amount of vaporized fuel entering the cylindres

135Using Fuzzy Logic forAutonomous Vehicle Motion Planning

Findings of Stanford Research Institute (SRI)

Based on the performance of the robot Flakey circa 1993

Discussion of autonomous navigation and path planning in an uncertain environment

Paper Using Fuzzy Logic for Autonomous Vehicle Motion Planning

136Difficulties of this problemFlakey

Autonomous operation of a mobile robot in a real-world unstructured environment poses a series of problems

knowledge about the environment is usually

incomplete

uncertain, and

approximate

Perceptually acquired information is not reliable

noise introduces uncertainty and imprecision

limited range and visibility introduces incompleteness

errors in interpretation

137More Difficulties with this Problem Flakey

Real world environments have complex and largely unpredictable dynamics

objects can move

the environment may be modified

features may change

Vehicle action execution is not reliable

the results produced by sending a given command to an effector can only be approximately estimated

action execution may fail entirely

138Robot Architecture using Fuzzy ControllerFlakeyMap of the roomsLPS

Key is Local

Perceptual

Space

LPS is data structure

providing

geometric

picture around

vehicle

Camera,etc 139The Fuzzy ControllerFlakey

Physical motion based on complex fuzzy controller

Provides a layer of robust high-level motor skills.

Basic building block of controller is a behavior

A behavior is defined as implementing an atomic motor skill aimed at achieving or maintaining a give goal situation

e.g. follow a wall.

140Implementing BehaviorsFlakey

Each behavioral skill is represented by means of a desirability function that expresses preferences over possible actions with reference to the goal

e.g. a behavior aimed at following a given wall prefers actions that keep the agent parallel to the wall at a safe distance

141Behavior through Fuzzy RulesFlakey

Each behavior was implemented by a set of fuzzy rules of the form

IF A THEN C

A is composed of fuzzy predicates and connectives, and

C is a fuzzy set of control vectors

An example of a keep off behavior rule is

IF obstacle-close-in-front AND NOT obstacle-close-on-left THEN turn-sharp-left

Last slide used 142Fast Reactive BehaviorsFlakey

Purely reactive behaviors, intended to provide quick simple reactions to potential dangers typically use sensor data that has undergone little or no interpretation.

Since quick response is necessary to avoid disaster, little processing can be done.

143Control StructuresFlakey

Purposeful behavior like attempting to reach a certain location must take explicit goals into consideration.

Goals represented in the LPS by means of control structures.

Control structure is a triple

S (A,B,C)

A is a virtual object (artifact) in the LPS

B is a behavior that specifies the way to react to the presence of this object, and

C is a fuzzy predicate expressing the context where the control structure is relevant

144Control Structure ExampleFlakeyS (A,B,C) A is a virtual object (artifact) in the LPS B is a behavior that specifies the way to react to the presence of this object, and C is a fuzzy predicate expressing the context where the control structure is relevant

An example control structure is

S1(CP1, go-to-CP, near(CP1)

CP1 is a control-point (marker for a location), together with a heading and a velocity

go-to-CP reacts to the presence of S1 in the LPS by generating the commands to reach the location, heading and velocity specified by CP1.

go-to-CP includes rules like

IF facing(CP1) AND too-slow-for(CP1) THEN accelerate-smooth-positive

145Blending of BehaviorsFlakey

Many behaviors can be simultaneously active

Fuzzy controller selects the controls that best satisfy the active behaviors

Satisfaction is weighted by each behaviors relevance to the current situation.

e.g. cant follow a wall if there isnt one

Context dependent blending of behaviors is implemented by combining the output of all the behaviors using context rules

146Generating a planS (A,B,C) A is a virtual object (artifact) in the LPS B is a behavior that specifies the way to react to the presence of this object, and C is a fuzzy predicate expressing the context where the control structure is relevantFlakey

simple goal-regressing planner used

based on a topological map annotated with approximate measurements (no obstacles) working backwards from goal.

An example plan might be

S1 (Obstacle, keep-off, near(Obstacle))

S2 (Corr1, follow,near(obstacle) AND at(Corr2) AND

near(Corr2))

S3 (Corr2, follow,near(obstacle) AND at(Corr2) AND

near(Door5))

S4 (Door5, cross,near(Obstacle) AND near(Door5))

Control structure 147Executing the PlanFlakey

S1 (Obstacle, keep-off, near(Obstacle))

S2 (Corr1, follow,near(obstacle) AND at(Corr2) AND

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